Matrix concentration inequalities, intimately connected to the Non-Commutative Khintchine inequality, have been an important tool in both applied and pure mathematics. We study tensor versions of these inequalities, and establish non-asymptotic inequalities for the $\ell_p$ injective norm of random tensors with correlated entries. In certain regimes of $p$ and the tensor order, our tensor concentration inequalities are nearly optimal in their dimension dependencies. We illustrate our result with applications to problems including structured models of random tensors and matrices, tensor PCA, and connections to lower bounds in coding theory. Our techniques are based on covering number estimates as opposed to operator theoretic tools, which also provide a geometric proof of a weaker version of the Non-Commutative Khintchine inequality, motivated by a question of Talagrand.